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SINTEF Energy Research
SINTEF Energy Research
2012-01-26
TR A7186- Unrestricted
Report
Pre-Analysis of daily and hourly raingauge
data
Report for the NFR Energy Norway funded project "Utilisation of weather radar data in
atmospheric and hydrological models"
Author(s)
Jean-Marie Lepioufle
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Document history
VERSION DATE VERSION DESCRIPTION
1 2012-01-26
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Table of contents
Part 1: daily raingauges network from eKlima, South-East part of Norway .................................................................................... 5
1 Introduction .................................................................................................................................................................................................................... 5
2 Theory background .................................................................................................................................................................................................... 5
2.1 Geostatistics, regionalized variables, and variogram. A brief recall ................................................................................5
2.1.1 Space-time values ...........................................................................................................................................................................5
2.1.2 Regionalized variables and random variables ..............................................................................................................6
2.1.3 Second order stationarity ...........................................................................................................................................................6
2.1.4 Intrinsic stationarity and variogram ....................................................................................................................................7
2.2 Geostatistical analysis tool : the climatological variogram cloud .....................................................................................8
2.3 Three intrinsic structures in a rainfall field .................................................................................................................................... 10
3 Raingauge network presentation ................................................................................................................................................................. 10
4 Pre-analysis of the spatial structure; a climatological variographical analysis ....................................................... 11
4.1 Second order and intrinsic stationarities verification ............................................................................................................ 11
4.1.1 Second order stationarity verification ............................................................................................................................ 11
4.1.2 Intrinsic stationarity verification ........................................................................................................................................ 14
4.1.3 Third chance to respect at least the intrinsic stationarity assumption : cutting the
space and the time domains ................................................................................................................................................ 14
4.2 Raingauge filtering using variographical threshold ................................................................................................................. 16
4.3 Influence of measurement .......................................................................................................................................................................... 20
4.3.1 Influence of the type of measurement ............................................................................................................................ 20
4.3.2 Influence of the quality of measurement ...................................................................................................................... 21
4.3.3 Influence of period of measurement ................................................................................................................................. 23
4.4 Presence of multiple independent processes with distinct direction .......................................................................... 26
4.5 Seasonal fluctuation influence ............................................................................................................................................................... 28
5 Atmospheric fluctuation influence ............................................................................................................................................................. 32
6 Discussion .................................................................................................................................................................................................................... 34
7 Conclusion .................................................................................................................................................................................................................... 34
Part 2: Hourly raingauges network eKlima, Orkla, Statkraft located over the Rissa radar area, Norway ............... 36
1 Introduction ................................................................................................................................................................................................................. 36
2 The network ................................................................................................................................................................................................................. 36
2.1 Homogeneity of the three datasets ..................................................................................................................................................... 38
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3 Overview of possible errors and/or possible presence of heterogeneous processes recorded by the different gauges ....................................................................................................................................................................................................... 38
3.1 Analysis of the statistical climatologic characteristic ........................................................................................................... 38
4 Conclusion .................................................................................................................................................................................................................... 42
1 References ................................................................................................................................................................................................................... 43
A.1 Test with the type measurement........................................................................................................................................................... 44
A.2 Effect of missing values on a structure ............................................................................................................................................ 46
A.3 Weather type decomposition, Tveito (2002) .................................................................................................................................. 47
A.4 Brief analysis of weather type time series ..................................................................................................................................... 49
A.5 Variogram clouds according to weather types ............................................................................................................................ 54
B Levene’s Test .............................................................................................................................................................................................................. 62
APPENDICES
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Part 1: daily raingauges network from eKlima, South-East part of
Norway
1 Introduction
With the objective of simulating realistic space-time rainfall over various catchments in Norway, we collected rainfall data from the eKlima database. Analyzing these data permits to determine input parameters in the stochastic rainfall generator. Norway is characterized by a severe and irregular topography. Rainfall process is strongly influenced by this particularity. By default, rainfall generator is able to simulate rainfall processes moving with an advection and characterized by stationary statistical characteristics in space and time. A pre-analysis step is necessary, firstly to verify the respect of the stationarity of the rainfall data, secondly to evince some mistakes and erroneous information. The pre-analysis is focused on geographical zone with relatively smooth topography. The region consists of Oslo, Akerhus, Østfold and Vestfold counties. Though a limited area, this technical report establishes an method that can be applied for the whole rainfall database. Part two will recall briefly the background theory used in this technical report. The rain gauge network will be presented in part three. The pre-analysis will be established in part four. Part five will present a brief analysis of using weather type for a possible improvement in the grouping similar rainfall field. Conclusions are given in the last part.
2 Theory background
Given that the main information input in the rainfall generator is a space-time structure model and its parameters, we focus this pre-analysis on the spatial variability. The analysis is based on geostatistics. A short summary of geostatistical theory is given in order to understand the different terms used in this technical report. This resume permits to present the different assumption used to model the spatial structure of a rainfall field and so to understand the different condition the data have to respect. For a more precise explanation of the theory, please refer to Journel and Huijbregts (1978) and Wackernagel (1995), Gneiting (2002). The spatial structure is determined by using the entire measuring period. It will give us the mean spatial structure, called the climatological spatial structure.
2.1 Geostatistics, regionalized variables, and variogram. A brief recall
Geostatistics gives the possibility to analyze and model the variability of environmental phenomena evolving in space and/or in time. The key terms are hereafter briefly introduced.
2.1.1 Space-time values
As many environmental variable, rainfall data are recorded in a space and time domain. It can be represented by a matrix of the form :
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, … , … ,
⋮ ⋮ ⋮, … , … ,
⋮ ⋮ ⋮, … , … ,
(2.1)
The variable z represents a rainfall measurement. Each row corresponds to an ensemble of values s recorded on a network of N-raingauges.. Each column corresponds to a time series of J-elements. Each variable , has a space-time coordinates , , . Two views can be used:
- To each location , corresponds a temporal rainfield.
- To each date corresponds a spatial rainfield.
In this resume we focus on the second view: a spatial rainfall field
2.1.2 Regionalized variables and random variables
Many natural phenomena are viewable to an ordinary observer as a regionalized shape evolving in a space or/and a time domain with some structure. Such phenomena can be characterized locally by magnitude varying in space or/and in time. In consequence, one can say that they constitute a numerical function called regionalized variables (ReV) ReV permits to easily represent natural phenomenon on a finite domain, but not to model it. A ReV process is realized using probabilistic theory and especially random function (RF) (Matheron, 1965). We assume that each measured value results in fact from an random mechanism. This mechanism is called random variable (RV). Thus, a measured value , , represents a random outcome from the RV , at time . From a probabilistic point of view, the ReV , , at location , , is one realization (one random) of a RV
, itself coming from a family of infinite RV, the random function , . The random function (RF) is characterized by distinct statistical and probabilistic characteristics. A rainfall process can therefore be represented by a ReV. And this ReV can be modeled by a RF.
2.1.3 Second order stationarity
One particular class of random function is the second order stationary RF. Stationarity of the second order signifies that the RF's statistical characteristics are identical and finite wherever the location in the domain. It is called the “invariance by translation”. In all point of space, we assume the following equality
- the mean :
(2.2)
- the variance :
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(2.3)
where is the spatial position , , is the expectation, and is the standard deviation.
Thus, the variability of two points distant by a vector , can be represented by a covariance which is independent of the location :
(2.4)
For a distance 0, the covariance 0 is equal to the variance .
2.1.4 Intrinsic stationarity and variogram
The second order stationarity is a RF's propriety and not a ReV's one. As precised by Wackernagel (1995), if in practice we call a “stationary ReV”, it is necessary to understand that this ReV can be represented by a realization of a second order stationary RF. We highlight here the fact that a second order stationary RF does not necessarily represent a ReV. The second order stationarity assumption can be sometimes too restrictive to represent observations. Indeed, an irregular mean and a non-finite variance can be observed in the ReV. Another assumption is introduced: the intrinsic stationarity of the zero order. This assumption doesn't compare anymore fixed point in space but the increment of two point separated by a vector
. Under a such assumption, the RF, also called RFI-0, respects the following proprieties: - the expectation of the increment is independent of the location and is equal to zero :
0
(2.5)
- the variance of the increment is only depending of vector :
2 (2.6)
These two RFI-0 proprieties lead to the variogram definition :
12
(2.7)
The variogram is a structure function use to model the variability (in space in our case) of a process. This function represents the background of the rainfall generator. If respects the second order stationarity, a link can be establish between the variogram and the covariance :
0 (2.8)
This case is illustrated on the figure 2.1
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The sill value 0 represents the a priori variance of the RF. When tending towards an infinity distance, the variogram reaches the a priori variance:
∞ 0 (2.9)
At the distance origin, the variogram takes a zero value :
0 0 (2.10)
Figure 1: Covariance and variogram evolution (in one dimension) of a RF when second order stationarity is
respected.
2.2 Geostatistical analysis tool : the climatological variogram cloud
Let us have a rainfall process being modeled by a general random process respecting no particular spatial assumption. Each time step corresponds to one spatial realization of this random function. Assuming a infinity of realizations (i.e. time step), determination of variogram parameters is possible. Unfortunately, the number of realization is finite, and spatial values are discrete and irregular. Nonetheless, expressing a discrete mean feature of the spatial structure is possible by computing a climatological variogram cloud. A variogram cloud describes the ensemble of all quadratic difference of values relative to all pair of measuring points present in one network. Each difference value is then put in a graphic taking into account the distance between the two measuring points. A climatological variogram cloud signifies that a mean of the quadratic difference is computed over the entire time series. A climatological variogram cloud value between two points separated by a vector is determined by :
12
(2.11)
where J represents the number of realizations (the number of time steps in our case). The climatological variogram cloud is a spatial characteristic exploratory tool. It enables:
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- To highlight some erroneous value and to locate the deficient measuring point.
- To highlight a possible presence of several independent processes evolving on the same measuring
area.
- To highlight a possible privileged direction in the rainfall process spatial structure
- To verify the assumptions of the RF which is request to model the studied rainfall process
Climatological variogram cloud is drawn as a 1D spatial graphic where each variogram values is positioned in function of the distance between two measuring points. The direction between raingauge is not take into account. In order to apply some test, climatological variogram cloud can also be drawn as a 2D spatial map. It permits to distinguish privileged direction or as a 1D graphic to only take into account the variability in function of the distance between two points (as shown on figure 2.2). A necessary information, to verify the second order stationarity is to indicate the a priori variance, drawn here the horizontal line. The higher value of the a priori variance in comparison to the order of magnitude of the variogram values is not a mistake. We recall that the a priori variance is calculated with all the values as expressed in (2.3). So this difference signifizes that with a daily resolution, the raingauges network doesn't catch the whole rainfall process.
Figure 2: Climatological variogram cloud where each circle represents the mean variability over the measuring period between two raingauges separated by distance km
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2.3 Three intrinsic structures in a rainfall field
Elaborating climatological variogram cloud is useful to diagnose the different question quoted in anterior part. For a more accurate diagnose, computing climatological variogram cloud is adaptable to other intrinsic structures linked to the rainfall field. These information are the non-zero rainfall structure, the rainy/non-rainy area transition structure and the rainfall indicator structure. The non-zero rainfall structure describes inner variability within a rainfall event. Intermittency is not taken into account. The rainfall indicator structure represent the structure of rainfall occurrence. The rain/non-rainy area structure has different feature according to the rain process, especially between frontal event and convective event. These three intrinsic fields will help us to verify the respect of statistical stationarity.
3 Raingauge network presentation
Daily rainfall data from a 149-raingauge network are collected. The recording time series cover the period from 01/01/1970 to31/12/2008. Different raingauge types are present: V: rainfall observed on a weather station, with measurements 3 to 8 times a day. A: rainfall automatically recorded by the station (in most cases every hour) N: rainfall recorded most of the times 1 time a day, in the morning. P: rainfall recorded every minute M: maritime weather station W: referenced raingauge without any measurement type Z: non-referenced raingauge The Z-type has been added by us. The raingauge network is located on an area regrouping four counties: Oslo, Akerhus, Østfold and Vestfold. This area was chosen due to the relatively gentle topography. Rainfall data are characterized by missing values (figure 3.1) and permanent zero value. The raingauges linked to these values are evinced from the analysis. The filtered raingauges are precised with the boolean filter “MValPZer” described in the file “SENorwRainNetSynthesis.xls”. A total of 21 raingauges are filtered:14.0% of the total network.
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4 Pre-analysis of the spatial structure; a climatological variographical analysis
The pre-analysis enables to highlight some specific characteristics of the rainfall process : possible presence of independent processes statistically different over a raingauges network, a privileged direction of process or evolution of rainfall process according to season. The pre-analysis is necessary before setting rainfall input parameters from data. Afterward, it enables a non-erroneous rainfall field simulation.
4.1 Second order and intrinsic stationarities verification
The first verification to do is to analyze the possibility of using either a covariance or a variogram as a spatial tool. This verification is established by looking at the two climatological first moment of the non-zero rainfall and of the rainfall indicator. If the second order stationarity assumption is respected, the climatological mean and the climatological standard deviation of all raingauges will be fixed in space. Then, covariance could be used as spatial tool. If not, the intrinsic-stationarity is studied. The climatological mean of the difference between values of all pair of raingauges over the whole network is calculated. The intrinsic stationarity is verified if every climatological difference is equal to a zero-value. Then the variogram can be used. Given that we are in an experimental domain, a quasi-zero-value is sufficient.
4.1.1 Second order stationarity verification
The non-zero rainfall climatological mean is characterized by a range between 2.89 and 8.45 mm/day. The minimal mean value is recorded by the raingauge #19450 at location (585.8199;6639.408). The maximal mean value is recorded by the raingauge #27430 at location (578.2507;6553.848). Prima facie, the difference seems big, but when calculating the mean value and the standard deviation of all raingauges's mean, their low value (resp 5.277 mm/day and 0.97mm/day) gentles this affirmation. A map of the interpolated mean values is represented in figure 4.1a. A third order polynomial interpolation is used. This tool is an easy way
Figure 3.1: Presence and absence of data for each raingauge during the recording
period (01/01/1970 to 31/12/2008)
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to see a spatial tendency without knowing any spatial dependance between the value and also without knowing the presence of mistakes. Of course, absence of measuring point involve a dramatic increasing interpolation value, but our subjective mind can avoid to take this information into account. The non-zero rainfall standard deviation is characterized by a range between 4.25 and 10.16 mm/day. The minimal value is recorded by the raingauge #19450 at location (585.8199;6639.408).. The maximal mean value is recorded by the raingauge #27350 at location (573.2263;6566.735). When merging all the raingauge measurements, the global standard deviation is equal to 7.150mm/day. The standard deviation of all standard deviation value of each raingauge is equal to 1.28 mm/day. Although the two extrema are perhaps significant, the global tendency of the standard deviation and the map (figure 4.1b) of interpolated values show not a quasi-homogeneity but at least a smooth heterogeneity. Given that the rainfall indicator is a Bernoulli value, the variance and so the standard deviation is directly linked to the mean as follow:
1 (4.1)
where and are the standard deviation and the mean of the rainfall indicator. Thus, only the mean value will be computed. The rainfall indicator mean gives an interesting and more physically interpreting information: the percentage of rainfall occurrence. Rainfall indicator is characterized by a minimal value equal to 0.08/day for the raingauge #27270 located at (581.5579;6571.884). The maximal value is equal to 0.70/day for the raingauge #2650 at location (644.2633 ;6644.444). The mean of all rainfall indicator is equal to 0.46/day and the standard deviation of the mean of each raingauge is equal to 0.08. It signifies a very uniform rainfall indicator field. Due to a smooth heterogeneity of the two first moment of the non-zero rainfall and the rainfall indicator, and some extreme characteristic values, the second order stationarity can not be accepted. Using covariance as spatial tool and as spatial model is refuted. The intrinsic stationarity assumption is thus tested.
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Figure 4.1: a-Non-zero climatological mean, b- Non-zero climatological standard deviation, c- Rainfall indicator
climatological mean over the measurement network without any permanent zero value and missing value
raingauge.
c-
b- a-
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4.1.2 Intrinsic stationarity verification
The climatological difference between values of each raingauge pair of the network are illustrated as a map where the x-axis and the y-axis represent the lag distance between pairs of raingauge. The map values have the particularity to be symmetric from the center of the map up to sign (figures 4.2b 4.2c). The uniformity of value locations enables a robust result (figure 4.2a). Both non-zero rainfall and rainfall indicator climatological difference are around the zero-value.(figures 4.2b 4.2c). But some extreme values disable the confirmation of a quasi-intrinsic stationarity. Non-zero climatological difference reaches an absolute extrema equal to 22.76 mm/day at interdistance location (33.00;33.91). Rainfall indicator difference reaches an absolute extreme equal to 0.37/day at interdistance location (7.45;84.75). Moreover, some sparse higher values can be seen on figures 4.3a and 4.3b representing climatological differences taking only into account the distance between raingauges and no more the direction.
4.1.3 Third chance to respect at least the intrinsic stationarity assumption : cutting the
space and the time domains
The intrinsic stationarity would be confirmed if no such high values were present. Only a few values seem to disturb the acceptance of the hypothesis. So, it will be interesting to isolate these raingauges responsible for high values and to verify if any mistake occurs, or if different statistical independent processes are present over a raingauges network, or if a privileged direction characterizes one or many processes. Using variogram as structure model is not yet confirmed, but using variogram cloud as a tool to study the spatial structure and the effect of each raingauge on the structure is powerful. It permits to highlight raingauges responsible of high values and then to filter them. But, filtering many raingauges because of their non-similar variability magnitude is not necessarily the best solution. Some external indicators will be used to cut the rainfall field in several fields respecting at least an intrinsic stationarity assumption. In fact, we already used an indicator to cut the rainfall field making it more homogeneous in the space domain. We use the topography irregularity cutting the rainfall field over the Norway territory to only keep a network over the four counties in the South-East of Norway. although the spatial stationarity is not respected, it would be worse taking into account the whole raingauge network. External indicators will be chosen as a spatial or/and temporal descriptor. One important remark is recalled. The aim, here, is only to cut a field in several pieces respecting intrinsic stationarity assumption. Indicators will not be used for the rainfall behaviour comprehension. This last sentence means that a model with a parameter similar to the indicator is already built. In this case, using an indicator will aim to verify if our rainfall model respects more or less the idea of the reality we expected. This is not a topic for this technical report. External indicators must be used to understand a process only if a rainfall model respecting more or less the idea of the reality we expected is already developed. This is not a topic for this technical report.
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Figure 4.2: a-Locations of each climatological difference between raingauge pairs, b- Non-zero climatological
difference map,c- Rainfall indicator climatological difference map of each raingauge pairs. Recall : raingauges
with permanent zero value and missing value are evinced from the study.
c-
b- a-
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Figure 4.3: a-Non-zero climatological difference and b- Rainfall indicator climatological difference of each
raingauge pairs as function of interdistance and not of the direction between each raingauge pair.
4.2 Raingauge filtering using variographical threshold
Drawing variogram clouds of total rainfall, transition rainfall, non-zero rainfall and rainfall indicator let appear high variographical values (figure 4.4a, b,c,d). Each circle corresponds to the climatological variability between two raingauges. Only a few and sparse values are high. A general tendency is not viewable. This values must be representative of mistakes and not natural effects. A threshold is choose to isolate this values and so to highlight raingauges behind these values. Of course the both raingauge of a pair are not responsible of this value. Only one can be responsible. And one raingauge can be responsible of several high pair values. The relative variance a priori value is drawn on the figure 4.4a, b,c,d.. This horizontal line is helpful in the choice of the threshold. Indeed, variographical values around this horizontal line are taken as plausible. The threshold value is always higher than this value. For information the variance of each the total rainfall field ans its intrinsic fields are wrote in table 1. The a priori variance of the transition variogram cloud is equal to :
2 (4.2)
b- a-
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Table 1: Variance values for the rainfall field and its intrinsic information
Field Total Transition Non-Zero Rainfall Indicator
Variance 30.635 (mm/day)� 39.485(mm/day)� 51.125(mm/day)� 0.249(./day)�
Filtering using variographical threshold has applied. Filtering has applied first to total rainfall, then to rainfall transition, third to non-zero rainfall, and finally to rainfall indicator variogram clouds.
Figure 4.4: a- Total rainfall variogram cloud, b- Rain/No-Rain transition variogram cloud, c- Non-zero rainfall
variogram cloud, d- Rainfall indicator variogram cloud. Recall : raingauges with permanent zero value and
missing value are evinced from the study.
d- c-
b- a-
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A total of 50 raingauges are filtered (33.4% of total raingauges). The filter list is explicated as a boolean list in the file “SENorwRainNetSynthesis.xls” in the column named “VariogThres”. As an indication, the percentage of filtered raingauges within each different types of measurement tool are wrote in the table 2.
Table 2: Percentage of filtered raingauge within each type of station measurement.
Station measurement
type
A N W Z
% of filtered raingauges 83 39 45 40
The locations of raingauges filtered by variographical threshold are illustrated in figure 4.5a. The locations of all filtered raingauges are illustrated in figure 4.5b. A general filter is described by the filter list in the file “SENorwRainNetSynthesis.xls” in the column named “FinalFilter”. The resulting variogram clouds are illustrated in figure 4.6 for the different intrinsic rainfall fields. For information,the variance value of the total rainfall and its intrinsic fields are give in table 3. Filtering raingauges doesn't alter the global signal. Even if high variogram values are deleted, this is normal to see variance value increases ; the variance is calculated with all measuring values but not with difference values. Deleting raingauges signifies increasing the variability between measurements.
Table 3: Variance values for the rainfall field and its intrinsic information
Field Total Transition Non-Zero Indicator
Variance 30.948 (mm/day)� 39.713(mm/day)� 51.348(mm/day)� 0.249(./day)�
In the figure 4.6, nice features appear without any sparse and high value anymore. The objective of filtering mistakes values is achieved For the total rainfall variogram, the structure seems to reach the variance (the sill) far after 200km. For the non-zero rainfall variogram, the increasing is faster. The non-zero rainfall variogram, also called inner variability, is coherent with clusters observable in a rainfall cell. In the case of the rainfall indicator variogram, the increasing toward the sill is slower. In extrapolating the variogram cloud, it seems to reach the variance between 800km and 1000km. The order of the distance which the variogram reaches the variance (the range) indicates we are studied frontal rainfall phenomena. Except a few values with high variability, the transition variogram cloud is also characterized by a long range. Going from a rainy to a non-rainy area is very smooth. It is also relative to a frontal process. Some remarks are however to be noticed. The few spread transition variogram values around the variance horizontal line signifies that for some raingauge pairs, transition between rainy and non-rainy area is abrupt, whatever short or long distance. This characteristic could be related to convective rainfall phenomena. In the case of the rainfall indicator variogram cloud, some spread values make the variogram cloud noisy.
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Moreover, some values are null. Is it only due to missing value then disturbing the variability or is it due to presence of a second or more rainfall process on the raingauge network? Some tests will be established later for rainfall indicator variogram and transition variogram to better understand these particularities. Of course, in our case studies, the rainfall signal and all the intrinsic information are very clear, and the above remarks are only light weighted details in the description. But this report describes also a method which can be applied subsequently for other raingauges network. The next parts will then focus on verifying homogeneity between measuring tools, on verifying rainfall process spatial stationarity for different direction and period during the year, and studying effect of missing values on spatial variogram cloud. In our case, higher variability is only a particularity of rainfall indicator and transition variogram clouds. Thus, tests will be established only on these two intrinsic fields. Of course, if particularity were present on other fields, such tests would be established on them.
Figure 4.5:Raingauge network with filter indication
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Figure 4.6: Variogram cloud taking account missing values and permanent zero filter and variographical
threshold filter : a-Total rainfall, b-Rain/no-rain transition c-Non-zero rainfall, d-Rainfall indicator
4.3 Influence of measurement
4.3.1 Influence of the type of measurement
The first possible heterogeneity can be due to non-similar measurement tools between the different types of weather stations. Conditioning each raingauges, variogram cloud relative to the type of measurement is computed (for rainfall indicator : figure 9.1, for rainfall transition : figure 9.2). Unfortunately, the number of
d c
b
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raingauges in each type is not equal and can be insufficient for a representative draw. It is possible to observe dense variogram cloud only for the type N and Z, where is a mix of all measurement tool untyped. So, no conclusion can be done with this test.
4.3.2 Influence of the quality of measurement
The second possible heterogeneity can be a consequence of the measurement quality. Indeed, measurement values are separated in nine different categories (table 4). In the study, a tenth category is added for values where no quality indication is present, this category takes the number -1. Two computing are established : one on data relative to “good” measurements, and a second to “slightly uncertain” measurements. The number of data in each group is noticed in table 5. Table 4: Quality control information of measurement data proposed in eKlima database
Indice Text Description
-2 Not in used -
0 OK Value is controlled and found O.K.
1 OK Value is controlled and corrected, or value is missing and interpolated manually.
2 Slightly uncertain Uncertain value is not controlled.
3 Slightly uncertain Uncertain reserved - not in use.
4 Slightly uncertain Uncertain value is slightly uncertain (not corrected).
5 Very uncertain Uncertain value is very uncertain (not corrected).
6 Very uncertain Uncertain, model data value is controlled and corrected, or value is missing and is
interpolated-automatic.
7 Erroneous Erroneous value (not corrected).
-1 No referenced No referenced
Table 5: Number of data according to the quality indicator
quality indicator 0 and 1 2, 3 and 4 0, 1, 2, 3, 4, 5, 6 and -1
number of data 319698 25862 705761
Rainfall indicator variograms and rainfall transition variograms are illustrated in figure 4.7 and 4.8. Lighter density of variogram relative to quality of measure 2,3 and 4 is due to less of disposable values.
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One remarks that the quality of measure is an indicator evolving in space and time, a contrario from the type of measurement which is only evolving in space. Exploring data with the quality indicator will increase missing values in the rainfall field. This effect could be viewable in aligned circle to the same variogram value. Some test will be done afterward to clarify impact of missing value. In conclusion, except to the cloud density, no major difference is notifiable between the two categories : the a priori variance are from the same order, so as the spatial structure.
Figure 4.7: Rainfall indicator variogram cloud according to the measurement quality: a-Categories #0 and #1, b-
Categories #2, #3 and #4
Figure 4.8: Transition rainfall variogram cloud according to the measurement quality : a-Categories #0 and #1,
b- Categories #2, #3 and #4
b- a-
b a
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4.3.3 Influence of period of measurement
The third possible heterogeneity is a possible change during the measurement period. Variogram clouds are computed for four different periods within the measurement time series : between January 1970 to December 1979, between January 1980 to December 1989, between January 1990 to December 1999, between January 2000 to December 2008. Rainfall indicator variogram cloud and rainfall transition variogram cloud are illustrated in figure 4.9 and figure 4.10. The major difference between the four sub-periods is the increasing variability in the fourth one. Prima facie, one think that the cause is a presence of a second independent rainfall process. The consequence is an apparition of a second structure formed by an ensemble of pairs values above the main structure. In comparison with figures 4.6b and 4.6d, the figures relatives to the year period between 2000 and 2008 are the most similar. If we base our reasoning on the first supposition, it signifies that a rainfall process appears between 2000 and 2008. It is not much to highlight a climate change. Unfortunately, the variogram cloud is a climatological one. Weight of years would have filtered new appearing process. It is important to remark that the number of missing values is more significant in the fourth sub-period (Table 6). It would explain the increasing variability and null values shown in figure 4.9 d. Indeed, if two raingauges are only represented with only a few of values in time, the probability it rains on the two raingauges is higher. However, a climatological variogram cloud would have annihilated these contrasted variogram values. In the figure 4.6b, zero values are still present. Thus, the presence of new raingauges in the fourth decade is responsible of this contrasting variability. Short time series according to these raingauges can't have climatological aspect. An illustration of erroneous in rainfall indicator variogram due to missing value is highlight in Appendix46. Assuming that these new raingauges detect the same process as the whole raingauge network and assuming that only a few raingauges are new. No influence of the period is concluded.
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Figure 4.9: Variogram cloud according to different period within the measurement time series: a-Between
January 1970 to December 1979, b- Between January 1980 to December 1989, c-Between January 1990 to
December 1999, d- Between January 2000 to December 2008
d c
b a
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Figure 4.10: Variogram cloud according to different period within the measurement time series: a-Between
January 1970 to December 1979, b- Between January 1980 to December 1989, c-Between January 1990 to
December 1999, d- Between January 2000 to December 2008
Table 6: Percentage of missing values within the four sub-periods of analysis
Year periode 1970-1979 1980-1989 1990-1999 2000-2008
% missing values 0.37 0.36 0.33 0.24
d- c-
b- a-
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4.4 Presence of multiple independent processes with distinct direction
We recall that this technical report is more a methodological report than a report focused on a specific database. With assumption that the recent raingauges detect the same rainfall process than the whole network, the above part conclude by no effect of period measurement. Take a more general supposition. Assume that we don't know if the new raingauges detect the same process. The aim of this part is to highlight the possible presence of a second detected process. This characteristic is represent in a double structure in the variogram cloud. In geostatistics, presence of double structure is called zonal anisotropy. It means that at least two independently process occur over an area where at least one has a different privileged direction. Assume that and represent variograms of two independent processes, where has no privileged direction and has a privileged direction. The consequence of a zonal anisotropy is illustrated in figure 4.11. For all direction different from , the spatial structure is characterized by . For the specific direction
, the spatial structure is the amount of the two independent variogram and . The rainfall indicator climatological variogram map is computed. To have continuous values, different interpolation method are tested. In a first step, polynomial interpolation of the third order is used (figure 4.12a). A privileged direction is pronounced around 80 degrees (trigonometric benchmark). But a contrario to the figure 4.11, the privileged direction is not linked to higher variogram values but to lower. We are in a case of a geometrical anisotropy which means that one process has a privileged directions: around 80 degrees. Although this interpolation tool is useful to represent the general feature of the process variability, it smooths the cover of the values. Using an interpolation method where variogram values are more represented and respected permits a better variability describing especially at small scale. As expected, with an inverse distance interpolation, the gentle geometrical anisotropy becomes more irregular (figure 4.12b). But no zonal anisotropy is detected. What is interesting to note is the evolution of the geometrical anisotropy with higher distance. Indeed the direction of the anisotropy tends to curve from 70 degrees to 90 degrees with increasing interdistance. A similar computation is done with non-zero rainfall (figure 4.12 c and d). No zonal anisotropy is detect in accordance the absence of double structure in figure 4.6 c. But a different privileged direction of the geometrical anisotropy is present. If not highlighting any zonal anisotropy, established variogram map on rainfall indicator and non-zero rainfall permits to show the usefulness of analyzing the rainfall field not as an isotropic process but as a process evolving differently in space according to its different intrinsic field. And explaining these behavior become another study: Is the geometrical anisotropy of the rainfall indicator is due to the shape of rainfall cell (elongated frontal cell), or is it due to an advection present at smaller time step. Similar question are possible for the shape of non-zero rainfall field. Are the cause similar to both intrinsic rainfall field. Which other parameters is able to interact with these field. Of course, this is not the aim of this report, but it permits to prevent the possible different process a raingauge network records and what we expect to simulate.
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b- a-
Figure 4.11: Zonal anisotropy
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Figure 4.12: Rainfall indicator climatological variogram map using different interpolation : a- polynomial of the
third order, b- inverse distance with a 10km resolution. Non-zero rainfall climatological variogram map using
different interpolation : c- polynomial of the third order, d- inverse distance with a 10km resolution,
4.5 Seasonal fluctuation influence
The objective is to isolate homogenous rainfall indicator processes, respecting at least intrinsic stationarity.. The above parts don't detect any climatological non-stationarity in the spatial domain. Our study was based on unchangeable process in time. However,rainfall process evolves throughout the year. When comparing maximum statistic difference during the year, change seems to appear. This evolution is described in the table 7 and the figures 4.13a and 4.13b. The mean and the standard deviation of the non-zero rainfall are higher between June and November. The coefficient of variation describes the dispersion of the values by the expression Cv=Standard deviation / mean. The higher the value, the bigger the dispersion. This coefficient looks mostly constant throughout the year. It indicates that non zero rainfall quantity evolve throughout the year but the rainfall process doesn't change. In the case of rainfall indicator, the higher values of the mean are present between September and January. A contrario of the non-zero rainfall the coefficient of variation of the rainfall indicator is not constant. The higher values are present between February and July. So rainfall indicator process evolves throughout the year.
d-
c-
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Table 7: Evolution of statistical climatological characteristics throughout the year of the non-zero rainfall and
the rainfall indicator
Non-zero rainfall Rainfall indicator
Month Mean
(mm/day)
Standard
deviation
(mm/day)
Coefficient of
variation
Mean (./day) Standard
deviation (./day)
Coefficient of
variation
January 4.329 5.661 1.308 0.518 0.250 0.482
February 3.961 5.405 1.365 0.436 0.246 0.564
March 4.506 5.843 1.297 0.43 0.245 0.570
April 4.27 5.546 1.299 0.398 0.240 0.602
May 4.926 5.996 1.217 0.383 0.236 0.617
June 5.595 7.025 1.256 0.442 0.247 0.558
July 5.984 8.058 1.347 0.445 0.247 0.555
August 6.113 8.599 1.407 0.468 0.249 0.532
September 6.268 8.486 1.354 0.496 0.250 0.504
October 6.355 8.628 1.358 0.541 0.248 0.459
November 6.079 7.811 1.285 0.545 0.248 0.455
December 4.597 6.039 1.314 0.496 0.250 0.504
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Figure 4.13: Evolution of statistical climatological characteristics throughout the year : a-Mean, standard
deviation and coefficient of variation of non-zero rainfall, b- Mean, standard deviation and coefficient of
variation of rainfall indicator
Evolution of statistical characteristics plays a role in the shape of the punctual distribution and also in the sill value of the variogram cloud. Evolution of a rainfall process is more linked to the evolution of the variogram cloud and especially to the value of the range. In the case of a frontal process, spatial variogram cloud increases slowly to the a priori variance. In opposition, a convective phenomenon describes a quick increase to the origin through the a priori variance in respect of the short spatial dimension. This behaviour can be viewable in the rainfall indicator variogram. Moreover, the a priori variance value relating to convective phenomena is higher than the frontal one. Indeed, convective rainfall process occur by little cells, in opposition with synoptic process such as frontal process resulting in uniform rain values in space. Convective rainfall process implies thus an higher variability and so an higher a priori variance value. This behaviour can be viewable in the non-zero rainfall variogram. Such statistical characteristics or processes evolving could be highlighted by coarsely cutting the year in two seasons. In our case, for example : winter and summer for the non-zero rainfall, and spring and fall for the rainfall indicator. Two rainfall indicator variogram cloud have been computed according to the spring and the fall season (figures 4.14a and 4.14 b). The two structures are almost similar. The a priori variance values are quasi-equal for both season (0.25/day). No structure reaches the sill further or nearer than the other. For both season, it seems that process with similar size occurs whenever in spring and in fall. In the case of non-zero rainfall, the variogram computed with summer values is linked by a priori variance higher than with winter values (63.146 mm²/day² in summer and 41.136 mm²/day² in winter). Transition variograms permits to highlight also the higher variability in summer when going from an rainy area to a non-rainy area in space (figure 4.16a and b).
b- a-
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In conclusion, we highlight that using seasonal cutting is not sufficient to isolate homogeneous rainfall process. A posteriori, we show an evolution of rainfall process throughout the year with a seasonal cutting, but presence of pure convective rainfall process is not highlighted. Obviously, this seasonal cut is very coarse. Rainfall process evolves from a season to another but also and especially from one day to another according to the atmospheric pattern. The next part focus on analyzing statistical characteristics of rainfall according the weather type furnished by the Norwegian Meteorological Institute.
Figure 4.14: Rainfall indicator variogram cloud taking account spring and fall season : a-Spring season (from
beginning March to end April), b-Fall season (from beginning September to end November)
b a
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Figure 4.15: Non-zero rainfall variogram cloud taking account summer and winter season : a-Summer season
(from beginning June to end August), b-Winter season (from beginning November to end February)
Figure 4.16: Transition variogram cloud taking account summer and winter season : a-Summer season (from
beginning June to end August), b-Winter season (from beginning November to end February)
5 Atmospheric fluctuation influence
Weather type decomposition give the opportunity to gather similar rainfall process without completely understanding the functioning of the different rainfall processes. Different weather type classifications exist e.g. Lamb (1972) who developed a subjective method, Jenkinson and Collison (1977) who used daily
b- a-
b- a-
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pressure at sea level to develop a semi-subjective method, or Paquer et al. (2006) who used daily precipitation to adapt weather type to rainfall. This technical report is based on weather type presented by Tveito (2002). A short description is given in Appendix1 A1.3. The Norwegian Meteorological Institute gives us a weather type indicator time series from the 1st of January 1971 to the 31th of July 2002. We analyze briefly the time series during this time period. Assuming that transition between every weather types are independent, we calculated the number of days, the number of events and the mean duration of one event according of each weather type. Figure 5.1 illustrates this information. The plot of number of days of each weather type is similar to the figure A1.5 in Appendix1 A1.3. To include seasonality in the study, similar calculations were done for to each month of the year (Appendix1 A1.4). Contrary to the anticyclonic weather type (#1), the cyclonic weather type (#10) and the weather types related to directional atmospheric flows (#19 to #26) are linked to seasonality. Hybrids weather types related to anticyclonic circulation describe also a light seasonality due to their coupling with direct flow circulations. In figure 5.1, duration of event of each weather type is rarely up to 2 days. Our model is based on cinematic of rainfall event. If setting all parameters from daily rainfall data where no cinematic is taken into account, our model becomes obsolete. Three alternatives are feasible. The first one is to base our study on hourly rainfall data. The second is to build weather type not only with see level pressure but also with rainfall value as established in Paquet et al (2006). The third possibility is to build no more weather type but rainfall type.
Nonetheless, a analysis of spatial variogram were established according to the weather type and for the different intrinsic rainfall fields. Given some preponderant weather types, only the types #1, #10 and #19 to #26 were taking into account. All these illustrations are presented in Appendix1 A1.5. No detailed
Figure 5.1: Monthly mean information of daily circulation pattern between the 1st of
January 1971 to the 31th of July 2002 : number of days, number of events, duration of
one event
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description is done. These figures are computed to highlight difference of rainfall variogram according to atmospheric fluctuation and to seasonality.
6 Discussion
Neither raingauge type nor quality flags for precipitation values play a major role in the non-respect of intrinsic stationarity. This non-respect is also not coming from presence of several rainfall processes on the study spatial area. Contrariwise, missing values alter dramatically the spatial variability information and especially the rainfall indicator variability. Thus a lot of raingauges have been filtered. Even if the number of raingauge is big enough, it would be interesting to achieve one method to take also into account the raingauges whose the time series is represented by a lot of missing values. No information would be lost. For example, a block variogram analysis method where each block would be inversely proportional to the number of missing values : the more missing values, the smaller the block. An important point would be to cover the entire area without superimposition of any blocks. Except for unrealistic variability of certain raingauge, spatial stationarity can be improved by cutting the measurement time series. A first seasonal analysis showed evolving non-zero rainfall statistical characteristics and also evolving rainfall indicator process throughout the year. An easy way to improve the gathering of rainfall process according to a spatial stationarity, is the use of daily classification with weather types as Tveito (2002). Variogram analysis of rainfall coupling with weather types enable to confirm rainfall field evolving according to atmospheric circulation Using weather type is an efficient way to gather rainfall process with similar statistical characteristics. However, the classification is based on a meteorological point of view but not on an hydrological one. It would be interesting to established some modifications in the classification construction: either in introducing some rainfall field characteristics in the atmospheric pattern determination as done by Paquet et al (2006), or in developing not a weather type but a rainfall type classification by using only rainfall characteristics (spatial and ponctual).
7 Conclusion
This report presents a method for data pre-analysis before setting parameters in our rainfall simulation. A specific area is chosen for spatial variogram analysis : South East of Norway. In a first part geostatistical concepts and used assumptions in the modeling are recalled. It permits to understand what for the pre-analysis is elaborate: to use our model, rainfall field has to respect at least the spatial intrinsic stationarity. Raw data don't respect this assumption. Therefore, in a second part, a pre-analysis protocol is established using variogram cloud thresholds to extract and then to oust responsible raingauges for unrealistic variability, and then using tools indicators (quality and type of measurement) to verify the spatial homogeneity of the measurements. Thirdly, external indicators (weather types) are used to enable a cutting of the rainfall field in group characterized by homogeneous statistics. After established this pre-analysis, one can use the rainfall data set to infer all the parameters needed in input of the rainfall generator.
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Each set of parameter enables a homogeneous rainfall simulation. To generate rainfall events necessitates to “paste” the different homogeneous rainfall simulations behind each other according to the time evolution of weather types (or other kind of type).
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Part 2: Hourly raingauges network eKlima, Orkla, Statkraft located
over the Rissa radar area, Norway
1 Introduction
The first has been more focused on the geostatitical analysis of the data set. In this second part, the analysis has been focused on the homogeneity of the variance (homogeneity of a population) and on classifying the raingauges according to their statistical characteristics.
2 The network
The raingauges network is composed of three datasets kindly provided by the Norwegian Meteorological Institute, and two private companies Orkla ASA and Statkraft AS. The dataset gathers 28 raingauges having recorded observation between 2006 and 2010 (5 years). Figure 1 shows the location of each raingauge over the Rissa radar area, area of interest, and Table 1 completes the coordinates and elevation information. Two raingauges have been evinced from the dataset at the beginning of the study due to permanent zero values in the time series: raingauges named 9580 and 63420. Figure1 highlights a strong presence of orography and also raingauges located at different elevation. Some network are preferentially located in higher elevation than other. This effect might have a consequence on the homogeneity of the statistical characteristics, and then heterogeneity of population. Total rainfall hasn’t been tested as such, but using a dichotomy delineating the rainfall indicator and the non-zero rainfall. This dichotomy enables to highlight easily different kinds of rainfall processes (convective, stratiform). We focused on the non-zero rainfall mean and variance and the rainfall indicator mean.
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Figure 3: location of the different raingauges used in the study. Statkraft (filled triangle), Orkla (filled circle), met.no (filled square), Rissa radar location (blank circle)
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2.1 Homogeneity of the three datasets
Because our hourly raingauge network is a gathering of three different network, it has been necessary to checkup the homoscedasticity (similar variance) for the three networks, by using the Levene’s test. Levene's test is an inferential statistic used to assess the equality of variances in different samples. The p-value from the F-distribution 100.4262,256992,2 provides the value 0.9900918 that confirms the hypothesis of homoscedasticity between the three datasets at 1% significance level.
3 Overview of possible errors and/or possible presence of heterogeneous processes
recorded by the different gauges
3.1 Analysis of the statistical climatologic characteristic
The analysis is first established using the climatologic statistical characteristics of the raingauges measurements. The values corresponding to each raingauge are referenced in table 2. Before starting classifying the raingauges, four others raingauges have been evinced from this part of the study due to a high amount of missing-values (61630, 67280, 67560, and 68860). The too few values within each raingauges don’t enable to get robust climatological characteristics. Nonetheless, it doesn’t mean yet those raingauges provide wrong values. They might be used for another purpose such as conditioning simulation. Two methods are used to classify raingauges into groups: a heuristic approach and the Self-Organizing Map (SOM) We used first a heuristic approach using only characteristics mostly used to detect presence of heterogeneous processes or errors: the non-zero rainfall variance, the rainfall indicator mean, the non-zero mean. In that case, five groups have been built following the idea that each group has to represent a homogeneous process. Each group is defined with qualitative indicators according to the statistical values of the entire dataset written in the table 2. Results of this classification are written in Table 3. In order to automatize the heuristic approach, a second method of classification has been used that is based on the artificial neuronal network: the SOM. It produces a two-dimensional, discretized representation of the input space of the training samples, called a map. This method has been developed by Teuvo Kohonen, and is therefore also called a Kohonen map (http://en.wikipedia.org/wiki/Self-organizing_map). Results are represented in Table 4 and are far from being different of the previous classification which is quite encouraging to provide automatic classification. This gauge classification enables to highlight different groups of raingauges within the network. Using climatologic characteristic, it doesn’t enable to point out some punctual measurement errors in time but recursive errors coming from one or several raingauges and/or different rainfall processes occurring over some measurement tools. In order to check-up whether these previous classification respect the homoscedasticity assumption, we use the Levene’s test for each of them. For both heuristic and Kohonen, the group 2 which represents raingauges with high variance doesn’t seem homogeneous, especially for the one coming from the Kohonen map (Table 6 and 7).
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Table 1: name, belonging, coordinates and elevation of each hourly raingauge of the network
# name dataset x(m utm32v) y(m utm32v) z (m) 1 9580 eklima 590 6910 482 2 61630 eklima 458 6900 579 3 63420 eklima 477 6950 6 4 67280 eklima 560 6980 299 5 67560 eklima 579 6980 127 6 68860 eklima 573 7030 127 7 69150 eklima 594 7040 40
8 BERKÅK orkla 552 6970 425 9 KVIKNE orkla 566 6940 550 10 LUSO orkla 575 7000 415 11 NERSKOGEN orkla 536 6970 665 12 SVARTELVA orkla 559 7060 124 13 SYRSTAD orkla 537 6990 125 14 YA orkla 580 6940 127 15 ØVREDØLVAD statkraft 560 6930 848 16 ÅMOTE statkraft 541 7000 227 17 HERSJØEN statkraft 610 7010 420 18 LØDØLJA statkraft 635 7000 540 19 NESJØEN statkraft 643 6990 725 20 SAKRISTIAN statkraft 673 6970 860 21 SELLISJØEN statkraft 636 6990 510 22 STUGUSJØEN statkraft 643 6980 630 23 SYLSJØEN statkraft 660 6980 840 24 AURA_AURSJØEN statkraft 476 6920 873 25 AURA_EIKESDAL statkraft 459 6930 61 26 AURA_HÅKODALSELV statkraft 480 6920 913 27 SVOR_SOLÅSVATN statkraft 494 6990 341 28 TROL_GRÅSJØ statkraft 507 6980 476
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Table 2: parameters values for each of the raingauges
# var %ind mean skew shape scale lambda number global 0.703 0.311 0.383 6.261 0.208 1.837 0.079 824773 1 NA NA NA NA NA NA NA 2 0.150 0.238 0.293 4.446 0.572 0.511 0.167 501 3 NA NA NA NA NA NA NA 4 0.385 0.107 0.495 3.139 0.638 0.776 0.316 2273 5 0.597 0.146 0.474 11.19 0.376 1.258 0.178 3175 6 2.340 0.073 0.890 4.476 0.339 2.624 0.302 2739 7 0.877 0.358 0.439 16.94 0.220 1.996 0.096 13792 8 0.374 0.287 0.302 8.632 0.244 1.237 0.0741 10070 9 20.21 0.251 0.465 67.93 0.01 43.39 0.005 8827 10 0.731 0.245 0.501 9.883 0.343 1.459 0.172 8606 11 0.443 0.323 0.298 10.85 0.201 1.486 0.060 11340 12 0.754 0.384 0.522 4.140 0.361 1.445 0.188 13492 13 0.553 0.255 0.413 5.691 0.308 1.338 0.127 8960 14 0.381 0.343 0.227 15.27 0.135 1.680 0.030 12028 15 0.246 0.305 0.201 12.83 0.165 1.221 0.033 10699 16 0.453 0.309 0.356 6.409 0.280 1.272 0.099 10852 17 0.441 0.412 0.381 10.11 0.329 1.157 0.125 13984 18 0.467 0.293 0.452 5.615 0.437 1.032 0.198 10182 19 0.467 0.339 0.376 5.699 0.303 1.241 0.114 11678 20 4.997 0.263 0.311 75.16 0.019 16.057 0.006 8996 21 0.469 0.291 0.402 6.234 0.345 1.165 0.1393 10039 22 0.396 0.286 0.371 5.648 0.347 1.067 0.1292 9892 23 1.834 0.413 0.213 103.04 0.02 8.607 0.005 14079 24 6.426 0.292 0.340 83.75 0.018 18.88 0.006 8945 25 0.837 0.283 0.510 6.18 0.311 1.638 0.159 9736 26 0.743 0.665 0.209 21.61 0.058 3.552 0.012 11215 27 0.863 0.398 0.639 5.259 0.472 1.351 0.302 13478 28 0.579 0.538 0.395 6.261 0.270 1.464 0.107 17417
Table 3: Heuristic (with only the non-zero variance (var) and the mean and the rainfall indicator mean (ind)), the different groups with respective raingauges, and the referential to classify the raingauges
Group # var ind mean 1: 9 very high normal lightly high 2: 20, 23,24 high normal normal 3: 17,26,28 normal high normal 4: 7,10,12,25,27 lightly high normal high 5: 8,11,13,14,15,16,18,19,21,22 normal normal normal
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Table 4: Kohonen 3x2 (with only var, ind, mean), the different groups with respective raingauges, and the referential to classify the raingauges
Group # var ind mean 1: 9 19.34 0.253 0.457 2: 20,24 5.725 0.279 0.325 3: 26,28 0.661 0.601 0.302 4: 7,10,12,25,27 0.814 0.327 0.514 5: 23 1.725 0.411 0.259 6: 8,11,13,14,15,16,17,18,19,21,22 0.421 0.314 0.340
Table 5: Kohonen 3x2 (with all the statistic characteristics), the different groups with respective raingauges, and the referential to classify the raingauges
Group Raingauge# var ind mean skew shape scale lambda 1: 9,20,23,24 9.089 0.308 0.335 82.43 0.018 22.93 0. 005 2: 7,14 ,26 0.669 0.442 0.300 17.72 0.144 2.339 0.049 3: 12,27 0.820 0.393 0.592 4.819 0.428 1.388 0.257 4: 16,21,25,28 0.585 0.364 0.417 6.264 0.303 1.384 0.127 5: 8,10,11,15,17 0.438 0.312 0.333 10.36 0.255 1.301 0.091 6: 13,18,19,22 0.469 0.295 0.403 5.662 0.351 1.164 0.143 Table 6: p-value from the Levene test related to the heuristic sorting with three characteristics
Group Nzr Rainfall indicator p-value significance level p-value significance level 1: - - - - 2: 0.9257728 <10% 0.9989891 <1% 3: 0.9997372 <1% 0.999996 <1% 4: 0.9957372 <1% 0.9993623 <1% 5: 1 <1% 1 <1% Table 7: p-value from the Levene test related to Kohonen 3x2 mapping with three characteristics
Group Nzr Rainfall indicator p-value significance level p-value significance level 1: - - - - 2: 0.2206537 <80% 0.9038534 <10% 3: 0.9584924 <5% 0.9707996 <5% 4: 0.9998181 <1% 0.9998788 <1% 5: 0.9999984 <1% 1 <1% 6: 0.9998812 <1% 0.9997923 <1%
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Table 8: p-value from the Levene test related to Kohonen 3x2 mapping with seven characteristics
Group Nzr Rainfall indicator p-value significance level p-value significance level 1: 0.9789308 <5% 0.9999489 <1% 2: 0.9968211 <1% 0.9996661 <1% 3: 0.9253532 <10% 0.7906506 <25% 4: 0.9979687 <1% 0.9999872 <1% 5: 0.999974 <1% 0.9999948 <1% 6: 0.9921765 <1% 0.9995205 <1%
4 Conclusion
This second part show an interesting and effective tool to classify the different raingauges. The method can also be used as a tool to evince some potential erroneous gauges. An effective pre-analysis would be to establish a SOM and after to perform a complete geostatistical analysis.
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1 References
Gneiting, T., 2002, Nonseparable, stationary covariance functions for space-time data, Journal of the American Statistical Association, 97(458), 590-600
Journel, A.G. and Huijbregts, C.J., 1978, Mining Geostatistics, Academic Press, London, 600p. Matheron, G., 1965. Les Variables Régionalisées et leur estimation. Masson, Paris, 305p Wackernagel, H., 1995. Multivariate Geostatistics : an Introduction with Applications., Springer-Verlag,
Berlin, 256 p
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A.1 Test with the type measurement
d c
b a
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Figure A1.1: Rainfall indicator variogram cloud taking account the type measurement : a-A, b-N, c-V, d-W, e-Z
d- c-
b- a-
e
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Figure A1.2: Rainfall transition variogram cloud taking account the type measurement : a-A, b-N, c-V, d-W, e-Z
A.2 Effect of missing values on a structure
Looking at the rainfall indicator climatological variogram cloud, some raingauges have identical rain occurrence whenever in time : some values are null (figure 9.3). It could be possible for close raingauges but not for those far from the other. The list of the raingauges and similarities are shown in table 8. It would be interesting to dig in this direction to limit the bad effect of missing value on spatial structure, and to optimize the knolegde of data event if rare data in the time series.
Figure A1.3: Rainfall indicator variogram cloud taking
account missing values and permanent zero filter and
variographical threshold filter
e-
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Table A1.1: Similitude of occurrence between raingauges after filtering missing values and permanent zero
taking account the variographical threshold filter
raingauge
#
2110 2100 2170 4070 19490 27430 30240 30230 27045
2110 identic
2100 identic identic identic identic identic identic
2170 0 identic identic identic identic
4070 identic identic
19490 identic identic
27430 identic identic identic identic identic
30240 identic identic identic identic
30230 identic identic identic identic identic
27045 identic identic identic identic identic identic identic
A.3 Weather type decomposition, Tveito (2002)
The three following figures A1.4, A1.5, and A1.6 describe the weather type decomposition used in this technical report. They come from the report written in 2002 by Tveito.
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Figure A1.5: Frequency of daily circulation indexes
1979-1994 (from Tveito, 2002)
Figure A1.4: Nodes used in the
classifiaction of large-scale
atmospheric circulation(from Tveito,
2002)
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A.4 Brief analysis of weather type time series
Type #1 to #9
Figure A1.6: Objective Lamb classification (after Brifa, 1995 ; Chen, 1999 ; Jones et al.,
1993) (from Tveito, 2002)
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Figure A1.7: Number of days of each weather type according to the month
Type #19 to #26
Type #10 to #18
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Type #10 to #18
Type #1 to #9
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Figure A1.8: Number of events of each weather type according to the month
Type #1 to #9
Type #19 to #26
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Figure A1.9: Mean duration of one event of each weather type according to the month
Type #19 to #26
Type #10 to #18
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A.5 Variogram clouds according to weather types
Type 20 Type 19
Type 10 Type 1
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Type 24 Type 23
Type 22 Type 21
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Figure A1.10: Rainfall indicator variogram cloud taking account weather types
Type 10 Type 1
Type 26 Type 25
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Type 22 Type 21
Type 20 Type 19
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Figure A1.11: Non-zero rainfall variogram cloud taking account weather types
Type 26 Type 25
Type 24 Type 23
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Type 20 Type 19
Type 10 Type 1
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Type 24 Type 23
Type 22 Type 21
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Figure A1.12: Transition variogram cloud taking account weather types
Type 26 Type 25
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B Levene’s Test
Levene's test is an inferential statistic used to assess the equality of variances in different samples. Some common statistical procedures assume that variances of the populations from which different samples are drawn are equal. Levene's test assesses this assumption. It tests the null hypothesis that the population variances are equal (called homogeneity of variance). If the resulting p-value of Levene's test is less than some critical value (typically 0.05), the obtained differences in sample variances are unlikely to have occurred based on random sampling. Thus, the null hypothesis of equal variances is rejected and it is concluded that there is a difference between the variances in the population. Procedures which typically assume homogeneity of variance include analysis of variance and t-tests. One advantage of Levene's test is that it does not require normality of the underlying data. Levene's test is often used before a comparison of means. When Levene's test is significant, modified procedures are used that do not assume equality of variance. The test is defined as follows:
∑ ̿
1 ∑ ∑ ̅
(A2.1)
where is the result of the test, is the number of different groups to which the samples belong, is the total number of samples, is the number of samples in the group. And , where is the
value of the sample from the group, and is the mean of the group. The term ̿ represents the mean of all . The significance of is tested against , 1, where is a quantile of the -test distribution, with 1 and its degrees of freedom, and is the chosen level of significance (usually 0.05 or 0.01).
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